The question provided is:

The first and last terms of an arithmetic progression are 12 and 48, respectively. The sum of the first four terms is 57. Find the number of terms in the progression.

Solution:

Let the first term of the arithmetic progression (AP) be $a = 12$, the last term $l = 48$, and the common difference $d$.

Step 1: Using the formula for the sum of an arithmetic progression

The formula for the sum of the first $n$ terms of an arithmetic progression is:

$S_n = \frac{n}{2} \times (2a + (n - 1)d)$

Given that the sum of the first four terms is 57, we can write:

$S_4 = \frac{4}{2} \times (2a + 3d) = 57$

This simplifies to:

$2(2a + 3d) = 57$ $4a + 6d = 57$ Substitute $a = 12$:

$4(12) + 6d = 57$ $48 + 6d = 57$ $6d = 57 - 48 = 9$ $d = \frac{9}{6} = 1.5$

So, the common difference $d = 1.5$.

Step 2: Finding the number of terms in the progression

We are also told the last term $l = 48$. The formula for the $n$-th term of an arithmetic progression is:

$l = a + (n - 1)d$

Substitute the known values $l = 48$, $a = 12$, and $d = 1.5$:

$48 = 12 + (n - 1) \times 1.5$ $48 - 12 = (n - 1) \times 1.5$ $36 = (n - 1) \times 1.5$ $n - 1 = \frac{36}{1.5} = 24$ $n = 24 + 1 = 25$

Thus, the number of terms in the arithmetic progression is 25.

Would you like more details on any step?

Here are 5 related questions you might find interesting:

1. How do you calculate the sum of any arithmetic progression?
2. How would the problem change if the common difference was negative?
3. What is the general formula for the $n$-th term of an arithmetic progression?
4. Can you find the sum of the entire arithmetic progression given this data?
5. What are some applications of arithmetic progressions in real-life problems?

Tip: Always check if the sum formula for arithmetic progressions applies fully or if modifications are needed for partial sums.